Feedback controls to ensure global solutions and asymptotic stability of Markovian switching diffusion systems

Guangliang Zhao; Fuke Wu; George Yin

doi:10.3934/mcrf.2015.5.359

Article Contents

2015, Volume 5, Issue 2: 359-376. Doi: 10.3934/mcrf.2015.5.359

This issue Previous Article Global controllability and stabilizability of Kawahara equation on a periodic domain Next Article

Feedback controls to ensure global solutions and asymptotic stability of Markovian switching diffusion systems

1.
GE Global Research, 1 Research Circle, Niskayuna, NY 12309
2.
School of Mathematics and Statistics, Huazhong University of Science and Technology, Wuhan, Hubei 430074
3.
Department of Mathematics, Wayne State University, Detroit, Michigan 48202

Received: January 2014

Revised: February 2014

Published: June 2015

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Abstract

To treat networked systems involving uncertainty due to randomness with both continuous dynamics and discrete events, this paper focuses on diffusions modulated by a continuous-time Markov chain. In our paper [19], we considered ordinary differential equations with Markovian switching. This paper further treats more complex cases, namely, stochastic differential equations with Markovian switching. Our goal is to stabilize the systems under consideration. One of the difficulties is that the systems grow much faster than the allowable rates in the literature of stochastic differential equations. As a result, the underlying systems have finite explosion time. To overcome the difficulties, we develop feedback controls to extend the local solutions to global solutions and to stabilize the resulting systems. The feedback controls are Brownian type of perturbations. We establish the existence of global solution, prove the stability of the resulting systems, obtain boundedness in probability as $t\to\infty$, and provide sufficient conditions for almost sure stability. Then we present numerical examples to illustrate the main results.

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Mathematics Subject Classification: Primary: 60J60, 60J27, 93E15; Secondary: 93E03.

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